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Greatest Common Divisor Algorithm

Publishing Venue

IBM

Related People

Cuff, RN: AUTHOR

Abstract

The drawing shows a method for finding the greatest common divisor (GCD) of a set of positive integers. It does not involve decomposition into primes, and is largely independent of the size of the integers and their pairwise differences. It relies on the fact that the GCD of X and Y is also the GCD of Y and X - mY for any integer m.

Country

United States

Language

English (United States)

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Greatest Common Divisor Algorithm

The drawing shows a method for finding the greatest common divisor (GCD)
of a set of positive integers. It does not involve decomposition into primes, and is
largely independent of the size of the integers and their pairwise differences. It
relies on the fact that the GCD of X and Y is also the GCD of Y and X - mY for
any integer m.

Let the numbers be N(1), N(2),...,N(J). Initially (step 1) the first pair is
assigned to T and V. Step 2 assigns the larger to C and the smaller to A. Step 3
calculates the remainder of C/A. If the division was not exact (step 4), then (step 5) the two smallest values (the remainder and A) are substituted and step 3
repeated. Eventually an exact division is found, and the current value of A is the
GCD of the pair.

If the end of the input list is reached (step 6), or there is already two coprime
numbers the final GCD must be unity, then the process is complete. Otherwise
(step 7), the GCD of A must be found and the next input number, and so on.